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232 CHAPTER 12. CONCLUSIONS & DIRECTIONS FOR FURTHER RESEARCH
Furthermore, for lower complexities the required overhead for multiple processors
dominates the actual computational costs and therefore a parallel approach requires
more processing time than a single processor implementation.
Finally, Section 7.5 describes four experiments in which the newly developed and
implemented JDCOMM method is used to compute the global minima. The global
minima in this section are also computed by the JD method and by the SOSTOOLS
and GloptiPoly packages to be able to compare the performance and the efficiency
of the various methods. The result here is that the JDCOMM method has a superior
performance over the JD method and the SOSTOOLS and GloptiPoly methods for all
the four experiments. The JDCOMM method uses fewer matrix-vector operations and
therefore also needs less computation time.
The performance of the iterative eigenvalue solvers is only efficient when the parameter
settings are well chosen. They are important to ensure the convergence to
the desired eigenvalue. We have obtained additional insight on these settings. In
particular, it is required to threshold the imaginary part of the target to prevent
constant switching of target due to the approximations.
In Part III of this thesis the approach of [50] plays an important role. In that
article the H 2 model-order reduction problem from order N tot N − 1 is solved by
algebraic methods. By approaching the problem as a rational optimization problem it
is reformulated as the problem of finding solutions of a quadratic system of polynomial
equations. In Chapter 8 this approach is generalized and extended which results in a
joint framework in which to study the H 2 model-order reduction problem for various
co-orders k ≥ 1. This makes it possible to solve the H 2 model-order reduction problem
from order N to N − 1, N − 2, and N − 3.
For co-orders k ≥ 1 we also end up with a system of equations in k − 1 additional
real parameters. Furthermore, any feasible solution of this system of equations should
satisfy k − 1 additional linear constraints. For computing the real approximations
of order N − k, the algebraic Stetter-Möller matrix method for solving systems of
polynomial equations of Section 3.3 is used. This approach casts the problem into a
particular eigenvalue problem involving one or more matrices in the k −1 parameters.
This approach guarantees also in this application that the globally best approximation
to the given system is found (while typically many local optima exist). To select the
global best solution the new and useful result of Theorem 8.2 is used where we present
for the co-order k case the homogeneous polynomial of degree three which coincides
with the H 2 model-order reduction criterion for the difference between H(s) and G(s)
at the stationary points of the system of equations. This is a generalization of the
third order polynomial for the co-order k = 1 case presented in [50].
In Chapter 9 this approach for k = 1 leads to a large conventional eigenvalue
problem. The solutions of this eigenvalue problem lead to approximations of order
N − 1 as described in [50]. However, we improve the efficiency of this method by a
matrix-free implementation which is possible by using the nD-systems approach as
presented in Chapter 5 in combination with an iterative Jacobi–Davidson eigenvalue